We study a scenario where a monitor is interested in the freshest possible update from a remote sensor. The monitor also seeks to minimize the number of updates that exceed a certain freshness threshold, beyond which, the information is deemed to be too old. Previous work has presented results for First Come First Served (FCFS) systems. However, it has been shown that Last Come First Served (LCFS) with preemption is more effective in terms of average Age of Information (AoI); we therefore study an M/G/1 LCFS system with preemption. The generality of the busy time distribution gives the advantage of applicability on any distribution inside the model. For example, one can use a deterministic distribution to study a TDMA system, a gamma distribution to model a routing network, or a more complicated distribution to study a CSMA access scheme. We find a general procedure to derive the exact expression of the outage update probability -- i.e. the portion of time updates have information older than a certain threshold. We compare different busy time distributions to the ones already present in literature for equivalent FCFS systems, showing the benefit of using the former discipline. We further study how the variance of the busy time distribution affects the update outage probability. We find two instances of the busy time distribution, where at low thresholds and low loads, higher variance gives an advantage in terms of update outage probability. First, we compare the M/D/1 LCFS with preemption against the M/$\Gamma$/1 LCFS with preemption and let the variance of the busy time of the latter vary, while maintaining the same average busy time for both systems. We further compare various M/$H_2$/1 LCFS with preemption with different coefficient of variation and same expected value, thus covering a wider spectrum of variation of the busy time.